What are people who 'flee' called?

Despite how weird or unglamorous, the word you are looking for is actually fleer. My dead-tree (thus not easily linkable) American Heritage Dictionary specifically lists it as a noun form at the end of its entry for flee.

Often 'refugee' may be a good noun to use when describing people who are fleeing.

How about fugitive? Here is how Encyclopédie Universelle defines that term:

a person who is fleeing, from prosecution, intolerable circumstances, etc.; a runaway: a fugitive from justice; a fugitive from a dictatorial regime.

Perhaps you could use escapee. Most nouns that mean to flee will carry strong connotations of something or another; it's just the nature of the game.